Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394823 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2416 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
927 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 72 points:
(1,2,5,6)(3,7,4,8)(9,10)(11,23)(12,29)(13,35)(14,21)(15,20)(16,22)(17,26)(18,33)(19,36)(24,39)(25,54)(27,42)(28,41)(30,43)(31,44)(32,46)(34,45)(37,48)(38,47)(40,57)(49,52)(50,55)(51,69)(53,58)(56,66)(59,61)(60,65)(62,64)(63,68)(67,71)(70,72);(2,6)(7,8)(9,11,14,26,16,28,15,27)(10,17,20,23,22,42,21,41)(12,30,32,53,34,52,33,24)(13,25,36,55,38,57,37,56)(18,43,45,39,46,49,29,58)(19,40,35,50,48,54,47,66)(31,60,61,71,63,70,62,51)(44,67,64,65,68,69,59,72);(1,3,5,4)(2,7,6,8)(9,12,15,33,16,34,14,32)(10,18,21,29,22,46,20,45)(11,24,27,52,28,53,26,30)(13,31,37,62,38,63,36,61)(17,39,41,43,42,58,23,49)(19,44,47,59,48,68,35,64)(25,51,56,70,57,71,55,60)(40,65,66,67,54,72,50,69);(1,4,5,3)(2,8,6,7)(9,13)(10,19)(11,25)(12,31)(14,36)(15,37)(16,38)(17,40)(18,44)(20,35)(21,47)(22,48)(23,50)(24,51)(26,55)(27,56)(28,57)(29,59)(30,60)(32,61)(33,62)(34,63)(39,65)(41,66)(42,54)(43,67)(45,64)(46,68)(49,69)(52,70)(53,71)(58,72)Polycyclic presentation:
2^9|a^2=ef,b^2=fi,c^2=ei,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308386);